I have an Archimedean-like Spiral Antenna (2-18GHz) of 61mm diameter. It is usually assumed that the far field region of an antenna lies approximately at a distance R> 2D^2/lambda, where D is supposed to be "the largest dimension of the antenna". What does this mean in a spiral antenna? I would put the diameter, but the resulting R is 2.5cm@2GHz and 22.2cm@18GHz.

This doesn't sound very nice. In principle, D has to be greater than lambda for the equation to be applicable. Then, what dimension D should I consider in such an antenna?

You're calculations are correct and you are using the correct D for the far field equation: R > 2*D^2/lambda (eq 1)

However, the far field also requires that R >> lambda (eq 2). Usually, eq 1 is the limiting equation, so it is used. But in this case you the 2nd equation would be the determining one because D is so small. Hence, at 2 GHz the far field wouldn't begin until R is roughly 10 lambda, or 1.5 meters.

If you look up the fields from a short dipole in a EM textbook or antenna book, you will see the near fields (that die off as 1/R^3 or 1/R^2) are present "close" to the antenna (when R is comparable to lambda). Once R is about 10 lambda away, these fields are negligible to the radiated fields (that die off as 1/R, and the E- and H-fields are orthogonal). The R>10lambda rule gets you out of the near field.